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The Metatheory of First-Order Logic: a Contribution to a Defence Of The metatheory of first-order logic: a contribution to a defence of Principia Mathematica Stephen Boyce University of Sydney 29 November 2010 Abstract This paper presents an account of the first-order logic of Principia Mathematica and evidence that the system is superior to currently accepted classical rivals. A widely accepted view that Whitehead and Russell’s presentation of logic in Principia is essentially defective in departing from contemporary standards is considered. It is shown that the judgement is based on a number of arguments that do not withstand critical scrutiny. For example, the assumption that the metatheory of contemporary first-order logic may be made precise in the expected way (using a first-order set theory equivalent to NBG) is shown to be false; on pain of contradiction, there cannot exist any such domain of interpretation of NBG. An alternative view of first- order logic, derived from Principia, is then presented. It is shown that Principia avoids the problem just discussed, as the first-order fragment may be made precise under an interpretation of the full sys- tem. arXiv:1003.4483v5 [math.LO] 28 Nov 2010 1 Is there a logic of Principia? My primary aim in the following is to sketch an account of the first-order logic of Principia Mathematica. I begin by considering the claim that the Whitehead-Russell presentation is flawed in so far as modern standards for the characterisation of a system of logic are not adhered to; hence to recover a system of logic from Principia it is essential to consider a revised system that adheres to contemporary standards must be developed. I challenge this claim 1 below (§2.1) by sketching a proof that the metatheory of modern first-order logic cannot be made precise in the generally expected way; more precisely, I show that (Proposition 2.1) there is no interpretation of NBG set theory that expresses the metatheory of Mendelson’s pure predicate calculus PP ([23]) . In §2.1 I focus on a standard presentation of a contemporary first-order logic, rather than a modern reconstruction of the first-order fragment of Principia, in order to support the claim that the Whitehead-Russell account of first-order logic ought to be considered on its merit. As the Whitehead- Russell account has been dismissed by many highly regarded authors (§2) it is necessary to present evidence in support of this claim. Having presented such evidence, I then (§3) examine the Whitehead-Russell version of first- order logic. I focus on the question of how to make Principia’s notions of a proposition / propositional functions precise - as a widely accepted objection to Principia holds that some confusion (perhaps involving sloppiness about the use/mention distinction) is intrinsic to the Whitehead-Russell account (cf [26], [3]: 56, [18]: 259). The slogan version of my account is that the logic of Principia is a logic of propositional functions, the latter being neither open sentences nor the ’meanings’ attached to these under some interpretation(s) but rather a combination of the two. In §5 I sketch an extension of this account to the full system of Principia. For brevity throughout the following various well-known distinctions con- cerning formal systems are blurred where no confusion should result. For example, it is convenient in many contexts to speak of the symbols of a for- mal system when it is really instances of the symbols that are mentioned; similarly, it is often convenient to refer to the meaning of a sentence and so on, when it is really the meaning of the sentence in a specific context that is referred to. As such distinctions are well-known in the literature and noth- ing essential rides on them in what follows I will make no further mention of them herein. 2 The formalist drive-by In this section I wish to consider the following claim (hereon ’the formalist drive-by’): to recover a system of logic from Principia it is essential to aban- don the Whitehead-Russell presentation and construct an alternative system that adheres to contemporary standards. Urquhart, in describing the type theory of Principia, explicitly asserts this claim: ... it is better to abandon the original Whitehead-Russell pre- sentation of the ramified theory of types and to follow a modern 2 presentation. The original presentation in Principia Mathemat- ica is both imprecise and notationally clumsy. Above all, the original formulation is unsatisfactory because there is no precise presentation of the syntax of the system. [29]: 295 The formalist drive-by is asserted implicitly by some authors. G¨odel, for example, to establish his famous incompleteness theorems concerning systems such as Principia, explicitly examines the formalist system P rather than the system of logic that Whitehead and Russell actually present in Principia. G¨odel is keen in [11] to establish that the exhibited proof catches the system of Principia itself; not surprisingly therefore the differences between the two systems are asserted to be of no essential import: P is essentially the system obtained when the logic of PM is superposed upon the Peano axioms ... The introduction of the Peano axioms, as well as all other modifications introduced in the system PM, merely serves to simplify the proof and is dispensable in principle. [11]: 599 That G¨odel actually judged the formalist reconstruction of, for example, Principia’s inference rules to be an essential correction to the Whitehead- Russell presentation is however made clear elsewhere. In characterising Prin- cipia’s place within the development of logic more generally G¨odel states: It is to be regretted that this first comprehensive and thorough- going presentation of a mathematical logic and the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations (contained in *1 - *21 of Principia) that it presents in this respect a considerable step backwards as com- pared with Frege. What is missing, above all, is a precise state- ment of the syntax of the formalism. ([12]: 120, modified through interpolation ’[]’) G¨odel’s considered position is thus that to extract a system of logic from Principia the Whitehead-Russell presentation must be modified so that a sys- tem with a precisely characterised, formal syntax (and so on) is described. When implicit and explicit assertions of the formalist drive-by are grouped together, it is clear that this claim is widely endorsed by many highly re- garded commentators: [4], [6], [8], [13], [14], [15], [16], [19] [20], [21], [22], [24], [26]. In examining the formalist drive-by it is useful to consider the claim that Whitehead and Russell ought to provide a precise presentation of the 3 syntax of the system. That is, they ought to have provided a ’purely formal’ description of the system of Principia ’in abstraction from’ any interpretation of the system ([4]: 58); minimally, they should have characterised the ’syntax’ of Principia in the narrower sense of specifying the formation and inference rules of the system in such terms. To illustrate what is at issue, Whitehead and Russell state the principle of inference generally known as modus ponens (for the propositional fragment of Principia) thus: ’∗1·1. Anything implied by a true elementary proposition is true.’ ([31]: 98). Jung amongst others takes Whitehead and Russell’s primitive proposition ∗1·1 to reflect a failure to distinguish an object language under discussion (namely the ideal language of Principia Mathematica) from a metalanguage in which this object language is being described ([16]: 137-140). On this view, the inference rule in question should be formulated in the metalanguage in a form such as the following: ’for any formulae φ and ψ, from φ and pψ → ψq infer ψ,’ ([16]: 140, using corners for quasi-quotation and syntactical variables ranging over the well- formed formulas of the object language). I note for future reference that Jung’s objection to ∗1·1 assumes that propositions are, in an appropriate sense, Principia formulas. It will be seen below that if Principia propositions are not formulas but other kinds of entities then the formalists have not demonstrated an error in the Whitehead- Russell presentation, but rather have failed to address the system under dis- cussion. Before considering what Principia’s propositions might be however I consider two arguments in favour of the claim that the Whitehead-Russell presentation of Principia is defective in so far as it departs from the the the formalists approach. These arguments may be summarised thus: Effectiveness requirement The Whitehead-Russell presentation, to the extent that it departs from the formalist approach, fails to provide a notion of proof that is effective, and hence the ideal language of Principia is not adequate for the purpose of communication (c.f. [4]: 50-52; [11]: 616 ’Note added 28 August 1963’); Appropriate metatheory requirement A formalist reconstruction of the logic of Principia is to be preferred to the original Whitehead-Russell presentation since the metatheoretical properties of the former system may be precisely characterised for at least the first-order fragment, and moreover for this fragment the properties are well-known and appropri- ate - in brief, the theory is sound, complete, consistent and undecidable. Consider firstly the effectiveness requirement. This claim is not essential to the formalists’ position, as the idea of a ’formal system’ may be defined more broadly so as to avoid this requirement (e.g. [23]: 25). That is as 4 well for the formalist perspective, as the routine communication of working mathematicians presents strong prima facie evidence against this claim. To illustrate this point in various ways I will assume that the currently accepted metatheory of (classical) first-order logic and so on is essentially correct.
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